1 Two-Dimensional Parity Internet Checksum Use 1-dimensional - PDF document

Error Coding Parity 1-bit error detection with parity Transmission process may introduce errors into a   message. Add an extra bit to a code to ensure an even (odd) number  of 1s Single bit errors versus burst errors  Every code word has an even (odd) number of 1s  Detection:  Requires a convention that some messages are invalid  Hence requires extra bits  An (n,k) code has codewords of n bits with k data bits and  111 011 r = (n-k) redundant check bits 01 11 010 110 Parity Correction Valid  Encoding: code Forward error correction: many related code words map to words  00 White – invalid 000 10 100 the same data word (error) 001 101 Detect errors and retry transmission  Basic Concept: Voting Hamming Distance Hamming distance of two bit  1-bit error correction with voting 1 0 1 1 0  HD=2 strings = number of bit 1 1 0 1 0 Every codeword is transmitted n times  positions in which they differ. If the valid words of a code  have minimum Hamming distance D, then D-1 bit errors can be detected. 111 If the valid words of a code 011  Voting: 010 110 have minimum Hamming Valid White – correct to 1 code 0 1 distance D, then [(D-1)/2] bit words 000 Blue - correct to 0 100 errors can be corrected. 001 101 Digital Error Detection Examples Techniques Two-dimensional parity  Parity  Detects up to 3-bit errors  111 011 Good for burst errors  01 11 010 110 Parity IP checksum Valid  Encoding: code Simple addition  words Simple in software 00 10 White – invalid 000 100  (error) Used as backup to CRC 001 101  Cyclic Redundancy Check (CRC)  Powerful mathematics  Voting  Tricky in software, simple in hardware  Used in network adapter  111 011 Voting: 010 110 Valid White – correct to 1 code 0 1 words 000 Blue - correct to 0 100 001 101 1

Two-Dimensional Parity Internet Checksum Use 1-dimensional parity Idea   Add up all the words Add one bit to a 7-bit code to ensure an   Parity even/odd number of 1s Transmit the sum  Bits Internet Checksum Add 2nd dimension   0101001 1 Use 1’s complement addition on 16bit codewords Add an extra byte to frame   1101001 0 Example Bits are set to ensure even/odd number of   Codewords: -5 -3 1s in that position across all bytes in frame  Data 1011110 1 1’s complement binary: 1010 1100  Comments  1’s complement sum 1000  0001110 1 Catches all 1-, 2- and 3-bit and most 4-bit  Comments  errors 0110100 1 Small number of redundant bits  Easy to implement  1011111 0 Not very robust  1111011 0 Parity Byte Cyclic Redundancy Check IP Checksum (CRC) u_short cksum(u_short *buf, int count) { Goal  register u_long sum = 0; Maximize protection, Minimize extra bits  while (count--) { Idea  sum += *buf++; Add k bits of redundant data to an n-bit message  if (sum & 0xFFFF0000) { N-bit message is represented as a n-degree polynomial with each bit in  /* carry occurred, so wrap around */ the message being the corresponding coefficient in the polynomial sum &= 0xFFFF; Example  sum++; Message = 10011010  } Polynomial  } = 1 ∗ x 7 + 0 ∗ x 6 + 0 ∗ x 5 + 1 ∗ x 4 + 1 ∗ x 3 + 0 ∗ x 2 + 1 ∗ x + 0 return ~(sum & 0xFFFF); = x 7 + x 4 + x 3 + x } CRC CRC - Sender Steps Select a divisor polynomial C(x) with degree k   T(x) = M(x) by x k (zero extending)  Example with k = 3:  Find remainder, R(x), from T(x)/C(x)  C(x) = x 3 + x 2 + 1  P(x) = T(x) – R(x) ⇒ M(x) followed by R(x)  Transmit a polynomial P(x) that is evenly divisible  Example  by C(x) M(x) = 10011010 = x 7 + x 4 + x 3 + x  C(x) = 1101 = x 3 + x 2 + 1  P(x) = M(x) + k bits  T(x) = 10011010000  R(x) = 101  P(x) = 10011010101  2

CRC - Receiver CRC – Example Encoding C(x) = x 3 + x 2 + 1 = 1101 Generator Receive Polynomial P(x) + E(x)  M(x) = x 7 + x 4 + x 3 + x = 10011010 Message E(x) represents errors  1101 10011010000 Message plus k E(x) = 0, implies no errors  1101 zeros Divide (P(x) + E(x)) by C(x)  1001 k + 1 bit check 1101 If result = 0, either sequence c,  equivalent to a 1000 No errors (E(x) = 0, and P(x) is evenly divisible by C(x)) Result:  1101 degree-k (P(x) + E(x)) is exactly divisible by C(x), error will not be  polynomial 1011 detected Transmit message 1101 followed by 1100 1101 remainder: 1000 1101 Remainder 10011010101 101 m mod c CRC – Example Decoding – CRC – Example Decoding – No Errors with Errors C(x) = x 3 + x 2 + 1 = 1101 Generator C(x) = x 3 + x 2 + 1 = 1101 Generator P(x) = x 10 + x 7 + x 6 + x 4 + x 2 + 1 = 10011010101 Received Message P(x) = x 10 + x 7 + x 5 + x 4 + x 2 + 1 = 10010110101 Received Message Received 1101 10010110101 1101 10011010101 Received 1101 message 1101 message, no errors 1000 1001 k + 1 bit check Two bit errors k + 1 bit check 1101 1101 sequence c, sequence c, equivalent to a 1011 equivalent to a 1000 Result: Result: 1101 1101 degree-k degree-k polynomial 1101 polynomial 1011 CRC test is passed CRC test failed 1101 1101 0101 1100 1101 1101 1101 Remainder Remainder 0 m mod c m mod c CRC Error Detection CRC Error Detection Odd number of bit errors can be detected if C(x) contains the factor (x + 1) Properties   Characterize error as E(x)  Proof: Error detected unless C(x) divides E(x)  C(x) = (x + 1) C’(x)  ( i.e. , E(x) is a multiple of C(x))  => C(1) = 0 (  C(x) has an even number of terms) What errors can we detect?  P(x) = C(x) f(x) = (x + 1) C’(x) f(x) All single-bit errors, if x k and x 0 have non-zero coefficients   => P(1) = 0 (  P(x) has an even number of terms) All double-bit errors, if C(x) has at least three terms  E(x) has an odd number of terms (odd number of error bits)  All odd bit errors, if C(x) contains the factor (x + 1)   E(1) = 1 Any bursts of length < k, if C(x) includes a constant term => P(1) + E(1) = 0 + 1 = 1 …………………………….. (1)  Most bursts of length ≥ k  Assume E(x) cannot be detected by CRC with C(x)   P(x) + E(x) = C(x)g(x) => P(1) + E(1) = C(1)g(1) = 0 ………………………………(2) (1) contradicts (2) => E(x) must be detected by C(x) 3

CRC Error Detection Common Polynomials for C(x) Any error bursts of length < k will be detected if C(x) includes a  CRC C(x) constant term x 8 + x 2 + x 1 + 1 CRC-8 Proof: E(x) = x i (x k-1 + ... + 1), where i >= 0 CRC-10 x 10 + x 9 + x 5 + x 4 + x 1 + 1  C(x) = x f(x) + 1  x 12 + x 11 + x 3 + x 2 + x 1 + 1 CRC-12 No power of x can be factored out of C(x)  => C(x) is not a factor of x i …………………….(1) CRC-16 x 16 + x 15 + x 2 + 1 C(x) has a degree of k: it cannot be a factor of polynomial with  smaller degree (up to k-1) x 16 + x 12 + x 5 + 1 CRC-CCITT => C(x) is not a factor of x k-1 + ... + 1 …………………….(2) CRC-32 x 32 + x 26 + x 23 + x 22 + x 16 + x 12 + x 11 + x 10 + x 8 + x 7 + x 5 + x 4 + x 2 + x 1 + 1 (1) (2) => C(x) cannot be a factor of E(x) Cyclic Redundancy Codes Error Detection vs. Error (CRC) Correction Detection  Commonly used codes that have good error Pro: Overhead only on messages with errors   detection properties. Con: Cost in bandwidth and latency for retransmissions  Can catch many error combinations with a small  number or redundant bits Correction  Based on division of polynomials. Pro: Quick recovery   Con: Overhead on all messages Errors can be viewed as adding terms to the   polynomial What should we use? Should be unlikely that the division will still work   Correction if retransmission is too expensive Can be implemented very efficiently in   Correction if probability of errors is high hardware.  Examples:  CRC-32: Ethernet  CRC-8, CRC-10, CRC-32: ATM  4